Question

given the information in the diagram and the table, how long is the wire that supports the tower? diagram: right triangle with 45° angle, base labeled 120 ft, wire (hypotenuse), tower (vertical side); table: $sin 30^circ = \frac{1}{2}$, $sin 45^circ = \frac{sqrt{2}}{2}$, $sin 60^circ = \frac{sqrt{3}}{2}$; $cos 30^circ = \frac{sqrt{3}}{2}$, $cos 45^circ = \frac{sqrt{2}}{2}$, $cos 60^circ = \frac{1}{2}$; text: exact values of common trigonometric functions

Explanation:

Step1: Identify the triangle type

The diagram shows a right triangle with the adjacent side to the \(60^\circ\) angle being \(12\) ft (distance from the base of the tower to the wire's anchor), and the wire is the hypotenuse. We use the cosine function, where \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Apply the cosine formula

For \(\theta = 60^\circ\), adjacent \(= 12\) ft, let hypotenuse (wire length) be \(x\). So \(\cos60^\circ=\frac{12}{x}\). From the table, \(\cos60^\circ=\frac{1}{2}\). Substituting, \(\frac{1}{2}=\frac{12}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(x\times1 = 12\times2\), so \(x = 24\) ft.

Answer:

The length of the wire is \(24\) feet.